Vertex renormalization in weak decays of Cooper pairs and cooling compact stars

At temperatures below the critical temperature of superfluid phase transition baryonic matter emits neutrinos by breaking and recombination of Cooper pairs formed in the condensate. The strong interactions in the nuclear medium modify the weak interaction vertices and the associated neutrino loss rates. We study these modifications non-perturbatively by summing infinite series of particle-hole loops in the S-wave superfluid neutron matter. The pairing and particle-hole interactions in neutron matter are described in the framework of the BCS and Fermi-liquid theories derived from microscopic interactions. Consistent with the f-sum rule, the leading order contribution to the polarization tensor arises at O(q^2) in the small momentum transfer, q, expansion. The associated neutrino emission rate is parametrically suppressed compared to its one-loop counterpart by a factor of the order of 5\times 10^{-3}, the parameter being the baryon recoil in units of temperature.